Improving the Erdős–Ginzburg–Ziv theorem for some non-abelian groups

Abstract

Let G be a group of order m. Define s(G) to be the smallest value of t such that out of any t elements in G, there are m with product 1. The Erdős-Ginzburg-Ziv theorem gives the upper bound s(G) 2m − 1, and a lower bound is given by s(G) D(G) + m − 1, where D(G) is Davenport's constant. A conjecture by Zhuang and Gao [J.J. Zhuang, W.D. Gao, Erdős-Ginzburg-Ziv theorem for dihedral groups of large prime index, European J. Combin. 26 (2005) 1053-1059] asserts that s(G) = D(G) + m − 1, and Gao [W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100-103] has proven this for all abelian G.In this paper we verify the conjecture for a few classes of non-abelian groups: dihedral and dicyclic groups, and all non-abelian groups of order pq for p and q prime.

Extremal product-one free sequences in Dihedral and Dicyclic Groups

“…• D(C q ⋊ s C m ) = m + q − 1 where q ≥ 3 is a prime number and ord q (s) = m ≥ 2 (J. Bass, [2]). For most finite groups, the Davenport constant is not known.…”

“…The proof of Theorem 1.4 is split up into sections 4, 5 and 6, where we solve the extremal inverse zero-sum problem associated to Davenport constant for dicyclic groups of orders 4n, where n ≥ 4, n = 3 and n = 2, respectively. The case n ≥ 4 is a direct consequence of Theorem 1.3 and the case n = 3 follows from Theorem 1.3, but in the special case (2). The proof in the case n = 2 is done manually, using only Theorem 2.2 to reduce the number of cases, without using Theorem 1.3.…”

“…Let D 2n be the Dihedral Group of order 2n and Q 4n be the Dicyclic Group of order 4n. J. J. Zhuang and W. Gao [19] showed that D(D 2n ) = n + 1 and J. Bass [2] showed that D(Q 4n ) = 2n + 1. In this paper, we give explicit characterizations of all sequences S of G such that |S| = D(G) − 1 and S is free of subsequences whose product is 1, where G is equal to D 2n or Q 4n for some n.…”

“…Suppose that S is a sequence in Q 8 with 4 elements that is free of product-1 subsequences. We want to show that S has some of those forms given in item (2). For this, we consider some cases depending on the cardinality of S ∩ H Q : …”

The Erdős–Ginzburg–Ziv theorem for finite solvable groups

“…Proofs for parts (a), (b), (c) and (d) can be found in [9,7,10]. We will prove only the last statement here.…”